On games of chance

Abstract

Early research in probability was concerned with the combinations arising in games of chance. Among the infinite variety of these combinations, some are easy to calculate, while others cause more difficulty, and since the difficulties increase as the combinations become more complicated, curiosity and the desire to overcome these difficulties prompted mathematicians to improve more and more upon this kind of analysis [1]. We have already seen that the profits shown by a lottery can easily be determined by the theory of combinations. But it is more difficult to know how many draws are needed for one to be able to lay a bet of 1 to 1, for example, that all numbers will be drawn. If n is the number of numbers, r the number of numbers drawn on each draw, and i the unknown number of draws, the expression for the probability that all the numbers will be drawn will depend on the nth finite difference of the ith power of the product of r consecutive numbers [2]. When n is very large, it becomes impossible to find the value of i that makes this probability equal to 1/2, unless this {finite} difference is converted into a series that converges rapidly. This can be successfully carried out by the method indicated above for approximating functions of very large numbers {of observations}. Thus in a lottery [3] composed of 10,000 numbers, only one of which is drawn at a time, there is some disadvantage in a 1 to 1 bet that all the numbers will appear in 95,767 draws, and an advantage in laying the same bet on 95,768 draws.